Concerning the geometry of stochastic differential equations and stochastic flows

Following Le Jan and Watanabe we define a connection associated with a non-degenrrate diffusion operators. This connection is characterized here and shown to be the Levi-Civita connection for gradient systems. This both explains why such systems have useful properties and allows us to extend these properties to more general systems. Topics described here include: moment estimates for $T{\xi }_t$, a Weitzenb"ock formula for the generator of the semigroup on p-forms induced by the flow, a Bismut type formula for $d\log p_t$ in terms of an arbitrary metric connection, and a generalized Bochner vanishing theorem. A comprehensive theory on this, its generalization to semi-elliptic case and applications is published in the book On the geometry of diffusion operators and stochastic flows'. Related to this is also the book The geometry of filtering’. This article is easier to read.